1 Real and Complex Domains in School Mathematics and in Computer Algebra Systems Eno Tõnisson University of Tartu Estonia

2 Plan Introduction School CASs Teacher Summary

3 Motivation: Unexpected answers CASs –are capable of solving many (school mathematics) problems –mostly solve as used at school, –but there are still answers more or less unexpected for school. Unexpected answers –are not inevitably mathematically incorrect –but may simply accord with another standard. Correctness, Completeness, Compactness Main goal is not only to find errors/dissimilarities but to use them positively.

4 Calculation, simplification of expressions, solving equations Unexpected answers Infinities and indeterminates Real and complex numbers (CADGME, today) Branches (ICTMT8, in July 1) Equivalence in CASs and school

5 Questions for all areas What exact commands would be useful if we try to get more school-friendly answers? How much are the CASs adjustable? Are there any special packages? What do CASs need in order to give more school-friendly answers? Why do CASs solve the problems as they do? Are different standards used? Are these standards useful for the school? Would it be possible to integrate these approaches to school treatment? Would it be reasonable?

6 Real and Complex Domains Real Imaginary Complex Rather: Border or bridge between R  C Real and imaginary

7 School and CASs School (different countries, textbooks, teachers) –Estonian, English, Norwegian, Russian –Primary and secondary school Grades ??-12, –University (teacher training) –General (??), Specific CASs (different systems, versions) –Derive 6, Maple 8, Mathematica 4.2, MuPAD 3.1, TI-92+, TI-nspire (prototype) and WIRIS. –General (??), Specific

8 Number Domains at School The available number domain gradually extends for the students during their school time. –In many countries (incl. Estonia) N  Q +  Q  R (  C) –Systematic Changeover may be complicated –N  Q discrete  dense. Merenlouto. What is next? Students –(probably?) work by default in their largest number domain 3(x-1)-(x+5)=2(x-4)  0 = 0 The solution set of this equation is the entire set of numbers known to us, that is, the rational number set Q. –usually do not think about number domain

9 Domain is important The topic of number domains is certainly important – –there may be different transformation rules allowed or (x,y ≥ 0) (R/C, H. Aslaksen) –the solution sets may differ in different number domains. It is not possible in “real” school to find –Square root for a negative number –Logarithm if argument or base are negative –Arc sine and arc cosine if argument is less than -1 or greater than 1. Using complex domain allows these operations –In case of square root is (probably) told that “restriction will be removed later”.

10 Complex Numbers at schools The school curricula –in many countries normally do not include complex numbers –in other countries complex numbers are a part of the school curricula. Only some elementary properties and operations treated –Introduction in secondary school ??? (if at all) –College Algebra course –Intermediate Algebra course Equality, Addition, Subtraction, Multiplication, (Division) (CA Barnett, Ziegler) Traditional university course of (Introduction to) Complex Analysis –More thoroughly –Imaginary unit occurs not only in case of square root but also in case of logarithms, inverse trigonometric functions, etc.?? –hopefully passed by math teachers

11 CAS Use of a CAS in the learning process creates a necessity and provides a chance to treat real and complex number domains more thoroughly. Test problems that –don’t initially include imaginary numbers –the solutions where CASs "cross the border" of real number domain.

12 “Visibility” of domain C? “Visible” i or C – –The imaginary numbers may appear in solutions of equations (already in case of quadratic equation). solve(x 2 = -1)  i, -i MuPAD: solve(0*x=0,x)  “Invisible” C answers –CAS may provide a solution of equation that is real number but is not appropriate when operating with real numbers only. solve( )  -1 –Equivalence of expressions (Separate paper) Equivalences known in school may not hold in CAS because of use of complex numbers (Aslaksen) What is the least restrictive constraint to make a given expressions equivalent?

13 Expectedness There are examples that teachers (and students?) –expect visible square root related (e. g quadratic equations) –but some examples are less known (“hardly expected”) visible logarithm related: ln(-1)  πi exponential equations e x +1 = 0 trigonometry: arcsin(2.0)  i trigonometric equations sin(x)=2 invisible C answers –radical equations –logarithm equations –arcus equations arccos(2x)=arccos(x+2) solution 2

14 Default (current) domain; What is the default domain in CAS? User manual (not always very informative) By default –Maple, Mathematica, MuPAD – C –Derive – C/(R) (solve Complex/Real), –TI-92+, TI-nspire – C/R (Complex Format Real/Rectangle/Polar, csolve) –WIRIS – R How “complex”? Test, –may be more detailed

15 Square root Logarithmarcsin arccos Calculationln(-1)arcsin(2) Equation (visible i) x 2 = -1e x =-1sin(x)=2 Equation (invisible C) arccos(2x)= arccos(x+2) Equation 0x=0 (visible C) Test problems

16 Complex domain ln(-1) arcsin(2) x 2 = -1 e x =-1 sin(x)=2 arccos(2x)=arccos(x+2) All All except WIRIS All except WIRIS and MuPAD All except WIRIS and TI-s All except WIRIS, Branches in Derive, MuPAD, TI-s All except WIRIS, Numerically arcsin(2.0) in Maple, Mathematica, MuPAD All except WIRIS, Branches in Derive, MuPAD, TI-s Numerically sin(x)=2.0 in Maple, Mathematica, MuPAD

17 Controllableness How could one set the domain (  R)? There are differences in the operation of different CASs – –in determination of domain of the calculation result, the variable value, the equation (inequality) solution the entire process.

18 How to determine the to R DeriveMapleMath-caMuPADTI-92+ TI-nspire WIRIS calculation result Packace RealDomain Packace RealOnly Complex Format Real default equation solution Solution Real Domain Packace RealDomain Packace RealOnly assumesolvedefault entire process Packace RealDomain Packace RealOnly default Not complete Exceptions (e.g Maple logarithmic equations)

19 Technical approaches Special Commands (cSolve) Assumptions Menu ->mode Menu-> radio button (Derive, Solve) Packages

20 Teacher actions Possible plan –clarify how a particular CAS works on a particular problem In tables of this paper? Test (guide will be in paper) –decide Avoid such problem in using CAS Adjust CAS (if possible) Add explanations (which?) –Is explanation useful and meaningful for student? –Will the topic be treated later? Don’t explain –???

21 Explanation? Too mathematical? L. Euler 1746 Complex logarithm is multivalued.

22 4 x = 64

23 Summary School –Merenluoto and Lehtinen: ‘‘little attention is paid to the underlying general principles of the different number domains in the traditional curriculum’’. –School treats complex numbers slightly if at all Use of a CAS in the learning process –creates a necessity and provides a chance to treat more thoroughly. CASs –are different in default domain in determination of domain –attempt to comply with pure mathematics rather than school mathematics –relatively well-adjustable (Assumptions, RealDomain, RealOnly.) Teacher must –know how particular a CAS works on a particular problem –choose a proper action (avoid, adjust, explain, ??)

24 Other areas Unexpected answers Infinities and indeterminates Real and complex numbers (CADGME, today) Branches (ICTMT8, in July 1) Equivalence in CASs and school Restrictive constraints

25 Future Work Systems and inequalities Other CASs, versions … Related works? Suggestions?